### Abstract

Let L^{m,p}ℝ^{n}) be the Sobolev space of functions with m^{th} derivatives lying in L^{p}(ℝ^{n}). Assume that n< p < ∞. For E ⊂ ℝ^{n}, let L^{m,p}(E) denote the space of restrictions to E of functions in L^{m,p}ℝ^{n}). We show that there exists a bounded linear map T: L^{m,p}(E) → L^{m,p}ℝ^{n}) such that, for any f ∈ L^{m,p}(E), we have Tf = f on E. We also give a formula for the order of magnitude of {norm of matrix}f{norm of matrix}L^{m,p}(E) for a given f: E → ℝ when E is finite.

Original language | English (US) |
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Pages (from-to) | 69-145 |

Number of pages | 77 |

Journal | Journal of the American Mathematical Society |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Fefferman, C. L., Israel, A., & Luli, G. K. (2013). Sobolev extension by linear operators.

*Journal of the American Mathematical Society*,*27*(1), 69-145. https://doi.org/10.1090/S0894-0347-2013-00763-8